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# Range Reporting for Time Series via Rectangle Stabbing

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LIPIcs.SWAT.2024.15.pdf
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## Cite As

Lotte Blank and Anne Driemel. Range Reporting for Time Series via Rectangle Stabbing. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.15

## Abstract

We study the Fréchet queries problem. It is a data structure problem for range reporting, where we are given a set S of n polygonal curves and a distance threshold ρ. The data structure should support queries with a polygonal curve q for the elements of S, for which the continuous Fréchet distance to q is at most ρ. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves of constant complexity and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point q should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with size in 𝒪(n log^{t-1} n) and query time in 𝒪(log^{t-1} n+k), where k denotes the output size and t can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. Note that we omit factors depending on the complexity of the curves that do not depend on n. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter d = ⌊t/2⌋ can be applied to our problem via reduction.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Data structures design and analysis
##### Keywords
• Data Structures
• Fréchet distance
• Rectangle Stabbing
• Orthogonal Range Searching

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## References

1. P. Afshani, L. Arge, and K.G. Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In Proceedings of the 2012 Symposuim on Computational Geometry, pages 323-338, 2012. URL: https://doi.org/10.1145/2261250.2261299.
2. P. Afshani and A. Driemel. On the complexity of range searching among curves. In Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 898-917, 2018. URL: https://doi.org/10.1137/1.9781611975031.58.
3. H. Alt. The computational geometry of comparing shapes. In Efficient Algorithms: Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, pages 235-248. Springer Berlin Heidelberg, 2009. URL: https://doi.org/10.1007/978-3-642-03456-5_16.
4. H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry and Applications, 5(01& 02):75-91, 1995. URL: https://doi.org/10.1142/S0218195995000064.
5. K. Bringmann, A. Driemel, A. Nusser, and I. Psarros. Tight bounds for approximate near neighbor searching for time series under the Fréchet distance. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 517-550, 2022. URL: https://doi.org/10.1137/1.9781611977073.25.
6. B. Chazelle. Filtering search: a new approach to query-answering. SIAM Journal on Computing, 15(03):703-724, 1986. URL: https://doi.org/10.1137/0215051.
7. B. Chazelle. Lower bounds for orthogonal range searching: I. the reporting case. Journal of the ACM, 37(02):200-212, 1990. URL: https://doi.org/10.1145/77600.77614.
8. Siu-Wing Cheng and Haoqiang Huang. Solving Fréchet distance problems by algebraic geometric methods. ArXiv, abs/2308.14569, 2023. URL: https://doi.org/10.48550/arXiv.2308.14569.
9. Mark de Berg, Atlas F. Cook, and Joachim Gudmundsson. Fast Fréchet queries. Computational Geometry, 46(6):747-755, 2013. URL: https://doi.org/10.1016/j.comgeo.2012.11.006.
10. A. Driemel and I. Psarros. ANN for time series under the Fréchet distance. In Algorithms and Data Structures, pages 315-328, 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_23.
11. A. Driemel and F. Silvestri. Locality-sensitive hashing of curves. In 33rd International Symposium on Computational Geometry, volume 77, pages 37:1-37:16, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.37.
12. H. Edelsbrunner and H.A. Maurer. On the intersection of orthogonal objects. Information Processing Letters, 13(04):177-181, 1981. URL: https://doi.org/10.1016/0020-0190(81)90053-3.
13. A. Filtser, O. Filtser, and M.J. Katz. Approximate nearest neighbor for curves: simple, efficient, and deterministic. Algorithmica, 2022. URL: https://doi.org/10.1007/s00453-022-01080-1.
14. M. Jiang and B. Zhu Y. Xu. Protein structure-structure alignment with discrete Fréchet distance. Journal of Bioinformatics and Computational Biology, 06(01):51-64, 2008. URL: https://doi.org/10.1142/s0219720008003278.
15. W. Meulemans. Similarity measures and algorithms for cartographic schematization. PhD thesis, Technische Universiteit Eindhoven, 2014. URL: https://doi.org/10.6100/IR777493.
16. E. Sriraghavendra, K. Karthik, and C. Bhattacharyya. Fréchet distance based approach for searching online handwritten documents. In Ninth International Conference on Document Analysis and Recognition (ICDAR 2007), volume 1, pages 461-465, 2007. URL: https://doi.org/10.1109/ICDAR.2007.4378752.
17. K. Toohey and M. Duckham. Trajectory similarity measures. SIGSPATIAL Special, 7(1):43-50, 2015. URL: https://doi.org/10.1145/2782759.2782767.
18. C. Wenk, R. Salas, and D. Pfoser. Addressing the need for map-matching speed: localizing global curve-matching algorithms. In 18th International Conference on Scientific and Statistical Database Management (SSDBM'06), pages 379-388, 2006. URL: https://doi.org/10.1109/SSDBM.2006.11.
19. T. Wylie and B. Zhu. Protein chain pair simplification under the discrete Fréchet distance. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 10(6):1372-1383, 2013. URL: https://doi.org/10.1109/TCBB.2013.17.
20. Y. Zhu, J. Peng, H. Liu, and Z. Lan. Chapter 26 - Analysis of nonadiabatic molecular dynamics trajectories. In Quantum Chemistry in the Age of Machine Learning, pages 619-651. Elsevier, 2023. URL: https://doi.org/10.1016/B978-0-323-90049-2.00013-5.